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Journal of Differential Equations 293 (2021) 313–358 www.elsevier.com/locate/jde
Dynamical systems under random perturbations with fast switching and slow diffusion: Hyperbolic equilibria ✩,✩✩ and stable limit cycles
∗ Nguyen H. Du a, Alexandru Hening b, Dang H. Nguyen c, George Yin d, a Department of Mathematics, Mechanics and Informatics, Hanoi National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam b Department of Mathematics, Tufts University, Bromfield-Pearson Hall, 503 Boston Avenue, Medford, MA 02155, United States c Department of Mathematics, University of Alabama, 345 Gordon Palmer Hall, Box 870350, Tuscaloosa, AL 35487-0350, United States d Department of Mathematics, University of Connecticut, Storrs, CT 06269, United States Received 3 September 2019; revised 11 March 2021; accepted 9 May 2021
Abstract This work is devoted to the study of long-term qualitative behavior of randomly perturbed dynamical systems. The focus is on certain stochastic differential equations (SDE) with Markovian switching, when the switching is fast varying and the diffusion (white noise) is slowly changing. Consider the system
√ dXε,δ(t) = f(Xε,δ(t), αε(t))dt + δσ(Xε,δ(t), αε(t))dW(t), Xε,δ(0) = x, αε(0) = i,
ε where α (t) is a finite state Markov chain with irreducible generator Q = (qι). The relative chang- ing rates of the switching and the diffusion are highlighted by two small parameters ε and δ. Asso-
✩ The work of N.H. Du was supported in part by NAFOSTED n0 101.02 - 2011.21; the work of A. Hening was supported in part by the National Science Foundation under grant DMS-1853463; the work of D. Nguyen was supported in part by the National Science Foundation under grant DMS-1853467; the work of G. Yin was supported in part by the National Science Foundation under grant DMS-2114649. ✩✩ We thank an anonymous referee who read the original version of the manuscript and suggested the study of the more general setting of hyperbolic equilibria, which has significantly improved the paper. * Corresponding author. E-mail addresses: [email protected] (N.H. Du), [email protected] (A. Hening), [email protected] (D.H. Nguyen), [email protected] (G. Yin). https://doi.org/10.1016/j.jde.2021.05.032 0022-0396/© 2021 Elsevier Inc. All rights reserved. N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 ciated with the above stochastic differential equation, there is an averaged ordinary differential equa- tion (ODE)
dX(t) = f(X(t))dt, X(0) = x, · = m0 · where f() ι=1 f(, ι)νι and (ν1, ..., νm0 ) is the unique stationary distribution of the Markov chain with generator Q. Suppose that for each pair (ε, δ), the process has an invariant probability measure με,δ, and that the averaged ODE has a limit cycle in which there is an averaged occupation measure μ0 for the averaged equation. It is proved in this paper that under weak conditions, if f has finitely many stable or hyperbolic fixed points, then με,δ converges weakly to μ0 as ε → 0and δ → 0. Our results generalize to the setting where the switching process αε is state-dependent in that
ε ε ε,δ ε ε,δ P{α (t + ) = |α = ι,X (s), α (s), s ≤ t}=qι(X (t)) + o( ), ι =
d as long as the generator Q(·) = (qι(·)) is locally bounded, locally Lipschitz, and irreducible for all x ∈ R . Finally, we provide two examples in two and three dimensions to showcase our results. © 2021 Elsevier Inc. All rights reserved.
MSC: 34C05; 60H10; 92D25
Keywords: Random perturbation; Random dynamical system; Invariant probability measure; Hybrid diffusion; Rapid switching; Limit cycle
Contents
1. Introduction ...... 314 1.1. Assumptions and main results...... 317 1.2. An application of Theorem 1.1 ...... 320 1.3. Main ideas of proof of Theorem 1.1: a road map...... 322 2. Estimates of the first exit times...... 325 3. Proof of main result...... 334 4. Proof of Theorem 1.2 ...... 339 4.1. An example...... 350 4.2. Another example...... 352 Appendix A. Proofs of lemmas in Section 2 ...... 353 References...... 357
1. Introduction
Natural phenomena are almost always influenced by different types of random noise. In or- der to better understand the world around us, it is important to study random perturbations of dynamical systems. In the continuous dynamical systems setup, the focus then shifts from the study of the behavior of deterministic differential equations to that of differential equations with switching (piecewise deterministic Markov processes) or stochastic differential equations with switching. The long-term behavior of these systems can be analyzed by a careful study of the ergodic properties of the induced Markov processes.
314 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358
Quite often, the “white noise” in the system is small compared to the deterministic com- ponent. In such cases, one is usually interested in knowing how well the deterministic system approximates the stochastic one. It is common to model continuous-time phenomena by stochas- tic differential equations of the type √ dxδ(t) = f(xδ(t))dt + δσ(xδ(t))dW(t), (1.1) where f(·) and σ(·) are sufficiently smooth functions, W(·) is a standard m-dimensional Brow- nian motion, and δ>0is a small parameter. If we let δ → 0, one would expect that the solutions of (1.1) converge to that of a deterministic differential equation in an appropriate sense. Different aspects of the problem have been studied extensively starting with Freidlin and Wentzell [36,17], Fleming [16], Kifer [27] and Day [11]. If the process xδ(t) has a unique ergodic probability measure μδ for each δ>0 and the origin of the corresponding deterministic ODE
dx = f(x)dt, (1.2) is a globally asymptotic stable equilibrium point, Holland established in [21] asymptotic expan- sions of the expectation of the underlying functionals with respect to the unique ergodic proba- bility measures μδ. In addition, in [22], Holland considered the case when the ODE (1.2) has an δ asymptotically stable limit cycle and proved the weak convergence of the family (μ )δ>0 to the unique stationary distribution that is concentrated on the limit cycle of the process from (1.2). Our interest in the current work stems from applications in ecology. Quite often, one models the dynamics of populations with continuous-time processes. This way we implicitly assume that organisms can respond instantaneously to changes in the environment. However, in some cases the dynamics are better described by discrete-time models, in which demographic decisions are not made continuously. In order to model more complex systems, one has to analyze ‘hybrid’ systems where both continuous and discrete dynamics coexist. Such systems arise naturally in ecology, engineering, operations research, and physics as well as in emerging applications in wireless communications, internet traffic modeling, and financial engineering; see [38]for more references. Recently, there has been renewed interest in studying piecewise deterministic Markov pro- cesses (PDMP) [10]. One may describe a PDMP by the use of a two component process. The first component is a continuous state process represented by the solution of a deterministic dif- ferential equation, whereas the second component is a discrete event process taking values in a finite set. This discrete event process is often modeled as a continuous-time Markov chain with a finite state space. At any given instance, the Markov chain takes a value (say i in the state space), and the process sojourns in state i for a random duration. During this period, the continuous state follows the flow given by a differential equation associated with i. Then at a random instance, the discrete event switches to another state j = i. The Markov chain sojourns in j for a random dura- tion, during which, the continuous state follows another flow associated with the discrete state j. A careful study of such processes has recently led to a better understanding of predator-prey communities where the predator evolves much faster than the prey [9] and for a possible explana- tion of how the competitive exclusion principle from ecology, which states that multiple species competing for the same number of small resources cannot coexist, can be violated because of switching [6,20]. It is natural to study the stochastic differential equation (SDE) counter-part of PDMP, that is, SDEs with switching. Similarly to the piecewise deterministic Markov processes mentioned in
315 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 the previous paragraph, in this setting one follows a specific system of SDEs for a random time after which the discrete event switches to another state, and the process is governed by a different system of SDE. The resulting stochastic process has a discrete component (that switches among a finite number of discrete states) and a continuous component (the solution of SDE associate with each fixed discrete event state). We refer the reader to [38]for an introduction to SDEs with switching. Most of the work inspired by Freidlin and Wentzell has been concerned with local phenomena that involve the exit times and exit probabilities from neighborhoods of equilibria. One usually uses the theory of large deviations to analyze the exit problem from the domain of attraction of a stable equilibrium point. Nevertheless, in applications such as those arising in ecology, one faces even more complex situations such as the one treated in this paper, in which large deviation techniques are not applicable, and one needs to analyze the distributional scaling limits for the exit distributions [2]. There have been some previous important studies for multi-scale systems with fast and slow scales [14,15]. These previous papers have looked at large deviations in the related setting where one has a slow diffusion and the coefficients are fast oscillating. However, in contrast to our framework, the fast oscillations come from having 1 periodic coefficients and inclusion of a factor ε into the periodic component of the coefficients. The way the fast oscillations are introduced in these previous papers is similar to how it is done when one does stochastic homogenization. In the present paper, the switching comes from a discrete random process αε. In this paper, we consider dynamic systems represented by switching diffusions, in which the switching is rapidly varying whereas the diffusion is slowly changing. To be more precise, let ( , F, {Ft }, P) be a filtered probability space satisfying the usual conditions. Consider the ε,δ process (X )t≥0 defined by √ dXε,δ(t) = f(Xε,δ(t), αε(t))dt + δσ(Xε,δ(t), αε(t))dW(t), Xε,δ(0) = x, αε(0) = i, (1.3) where W(t)is an m-dimensional standard Brownian motion, αε(t) is a finite-state Markov chain that is independent of the Brownian motion and that has a state space M ={1, ..., m0} and ε,δ d d d d generator Q/ε = qij /ε , X is an R -valued process, f : R × M → R , σ : R × m0×m0 M → Rd×m, and ε, δ>0are two small parameters. We assume that the matrix Q is irreducible. The irreducibility of Q implies that the Markov chain associated with Q, denoted by (α(t))˜ t≥0, is ε,δ ergodic thus has a unique stationary distribution (ν1, ..., νm0 ). We denote by Xx,i(t) the solution ≥ ε of (1.3)at time t 0 when the initial value is (x, i) and by αi (t) the Markov chain started at i. Let us explore, intuitively, what happens when ε and δ are very small. In this setting, αε(t) converges rapidly to its stationary distribution (ν1, ..., νm0 ), while the diffusion is asymptotically small. As a result, on each finite time interval [0, T ] for T>0, a solution of equation (1.3) can be approximated by the solution Xx(t) to
dX(t) = f(X(t))dt, X(0) = x, (1.4) = m0 where f(x) i=1 f(x, i)νi . However, if in lieu of a finite time horizon, we look at the process on the infinite time horizon [0, ∞), it is not clear that Xx(t) is a good approximation. Suppose that equation (1.4) has a stable limit cycle. A natural question is whether the invariant probability measures (με,δ) of the processes (1.3) converge weakly as ε → 0 and δ → 0, to the measure concentrated on the limit cycle. This is the main problem that we address in the current paper. In order to do this, we
316 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 substantially extend the results of [22]by considering the presence of both small diffusion and rapid switching. Because of the presence of both the switching and the diffusion we need to develop new mathematical techniques. In addition, even if there is no switching and we are in the SDE setting of [22], our assumptions are weaker than those used in [22]. The rest of the paper is organized as follows. The main assumptions and results are given in Section 1.1. To provide an insight of the proofs and to connect different parts of the arguments so as to provide something like a “road map”, Section 1.3 presents the main ideas of proofs. In Section 2, we estimate the exit time of the solutions of (1.3) from neighborhoods around the stable manifolds of the equilibria of f . The proofs of the main results are presented in Section 3. In Section 4, we apply our results to a general predator-prey model. In addition to showcasing our result in a specific setting, the proofs in Section 4 are interesting on their own right as they are quite technical and require the development of new tools. Finally, in Section 4.1, we provide some numerical examples to illustrate our results from the predator-prey setting in Section 4.
1.1. Assumptions and main results
We denote by A the transpose of a matrix A, by | ·|the Euclidean norm of vectors in Rd , and by A := sup{|Ax| : x ∈ Rd , |x| = 1} the operator norm of a matrix A ∈ Rd×d . We also define a ∧ b := min{a, b}, a ∨ b := max{a, b}, and the closed ball of radius R>0 centered at the origin d BR := {x ∈ R :|x| ≤ R}. We recall some definitions due to Conley [8]. Suppose we are given a flow ( t (·))t∈R. A compact invariant set K is called isolated if there exists a neighborhood V of K such that K is the maximal compact invariant set in V . A collection of nonempty sets {M1, ..., Mk} is a Morse decomposition for a compact invariant set K if M1, ..., Mk are pairwise disjoint, compact, isolated sets for the flow restricted to K and the following properties hold: 1) For each x ∈ K there are integers l = l(x) ≤ m = m(x) for which the alpha limit set of x, ˆ = { ∈ −∞ ]} ˆ ⊂ ˆ := α(x) t≤0 s(x), s ( ,t , satisfies α(x) Ml and the omega limit set of x, ω(x) { ∈[ ∞ } ˆ ⊂ = ∈ = t≥0 s(x), s t, ) , satisfies ω(x) Mm 2) If l(x) m(x) then x Ml Mm.
Assumption 1.1. We impose the following assumptions for the processes modeled by systems (1.3) and (1.4).
(i) For each i ∈ M, f(·, i) and σ(·, i) are locally Lipschitz continuous. (ii) There is an a>0 and a twice continuously differentiable real-valued, nonnegative function (·) satisfying lim inf{ (x) :|x| ≥ R} =∞and (∇ ) (x)f (x, i) ≤ a( (x) + 1), for all R→∞ (x, i) ∈ Rd × M. · 1 { } (iii) The vector field f() is C and it has finitely many equilibrium points x1, ..., xn0−1 and a unique limit cycle . The equilibrium points are hyperbolic points. { ··· } (iv) There exists a Morse decomposition M1, M2, , Mn0 of the flow associated with f such = ={ } that Mn0 is the limit cycle and for any i
317 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358
Remark 1.1. We note that using Assumption (ii), we can work in a compact state space K ⊂ Rn if the diffusion term from (1.3)is zero. Assumptions (i) and (ii) are needed in order to deduce the existence and boundedness of a unique solution to equation (1.3)in the absence of the diffusion term. Assumption (iv) is used to make sure that there exist no heteroclinic cycles. Assumption (v) ensures that (1.3) has a unique solution that is strong Markov. Sufficient conditions that imply uniqueness and the strong Markov property exist in the literature [32,38].
Remark 1.2. In [22]the author studied (1.1) under the assumptions that
(A1) f, σ ∈ C2(Rd ). (A2) System (1.2) has a unique limit cycle. (A3) There exists at most a finite number of equilibria x∗ of f . At each equilibrium, the Jacobian matrix has only positive real parts and the matrix σ σ is positive definite. (A4) For any compact set B not containing equilibria and any u > 0 there exists T>0 such that if x ∈ B, then
0 ≥ d(xx (t), ) < u, for t T.
(A5) There exists δ0 > 0 such that for 0 <δ<δ0 the stochastic differential equation has a unique d ergodic measure μδ. Furthermore, the family (μδ)0<δ<δ0 is tight in R .
Our work generalizes [22] significantly in the following aspects. First, we work with two types of randomness - one comes from the diffusion term and the other from the switching mechanism. Second, Assumption 1.1 (i) is weaker than (A1). Third, we can have any hyperbolic fixed points whereas assumptions (A3)-(A4) imply that all fixed points are sources and the deterministic system converges uniformly to the limit cycle. In addition, we do not need σ σ to be positive definite at the equilibria.
Remark 1.3. There are several papers, which look at the exit time asymptotic behavior near hyperbolic fixed points of small perturbations of dynamical systems [27,1,2]. In contrast to these papers in which the noise is uniformly elliptic, we have to deal with the additional complications of a stable limit cycle as well as the switching due to αε.
0 Let T > 0be the period of the limit cycle . We can define a probability measure μ , which is independent of the starting point y ∈ , by
T 0 1 μ (·) = 1(·)(Xy(s))ds, (1.5) T 0 where Xy(t) is the solution to equation (1.4) starting at X(0) = y and 1{·} is the indicator function. The measure μ0(·) is the averaged occupation measure of the process X restricted to the limit cycle . Throughout the paper, we assume that δ depends on ε, i.e., δ = δ(ε), and lim δ(ε) = 0. We will investigate the asymptotic behavior of the invariant probability measures ε→0 με,δ as ε → 0in the following three cases:
318 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 ⎧ ⎨ l ∈ ( , ∞), δ 0 case 1 lim = 0, case 2 (1.6) → ⎩ ε 0 ε ∞, case 3.
The multi-scale modeling approach we use is similar to the one from [23].
Assumption 1.2. We impose additional conditions corresponding to the cases from (1.6).
δ = ∈ ∞ ∗ ∗ ∈ M 1) Suppose lim→0 ε l (0, ). For any equilibrium x of f there exists i such that β f(x∗, i∗) = 0 or β σ(x∗, i∗)σ (x∗, i∗)β = 0 where β is a normal vector of the stable man- ifold of (1.4) at x∗. δ = ∗ ∗ ∈ M 2) Suppose lim→0 ε 0. For any equilibrium x of f there exists i such that β f(x∗, i∗) = 0 where β is a normal vector of the stable manifold of (1.4) at x∗. δ =∞ ∗ ∗ ∈ M 3) Suppose lim→0 ε . For any equilibrium x of f , there exists i such that β σ(x∗, i∗)σ (x∗, i∗)β = 0 where β is a normal vector of the stable manifold of (1.4) at x∗.
Note that when x∗ is a source, the stable manifold is 0-dimensional and any non-zero vector can be considered as a normal vector.
The intuition for the conditions of Assumption 1.2 is as follows. In case 2, since δ tends to 0 much faster than ε, for sufficiently small δ, the behavior of Xε,δ(t) will be close to the process ξ ε(t) defined by dξε(t) = f ξ ε(t), αε(t) dt. (1.7)
ε ≥ We denote from now on by ξx,i(t) the solution of (1.7)at time t 0if the initial condition is (x, i). ∗ ∗ ∗ If for each i ∈ M, f(x , i) = 0at a equilibrium x of f , the Dirac mass function at x , δx∗ , will be an invariant measure for ξ ε(t). Because of this, the sequence of invariant probability ε,δ measures (μ ) (or one of its subsequences) may converge to δx∗ . In order to have the weak δ,ε 0 ∗ convergence of (μ )ε>0 to the measure μ , we need to assume that there is an i ∈ M such that β f(x∗, i∗) = 0 where β is a normal vector of the stable manifold of (1.4)at x∗. This guarantees that the process from (1.7) gets pushed away from the equilibrium x∗ and away from the stable manifold (where it could get pushed back towards the equilibrium). In case 3, the switching is very fast compared to the diffusion term, so for small ε the process will behave like √ dηε(t) = f(ηε(t))dt + δσ(ηε(t))dW(t),
1 = 2 where σ(x) i∈M σ(x,i)σ (x, i)νi and W(t)is an independent n-dimensional Brownian motion. In order for the limit of (με,δ) not to put mass on the equilibrium x∗ of f , we need to suppose that there exists an i∗ ∈ M such that β σ(x∗, i∗) = 0 where β is a normal vector of the stable manifold of (1.4)at x∗.
319 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358
For case 1, since both the switching and the diffusion are on a similar scale, we need to assume that for each equilibrium x∗ of f there is i∗ ∈ M satisfying either β σ(x∗, i∗) = 0or β f(x∗, i∗) = 0. The next theorem is the main result of this paper.
Theorem 1.1. Suppose Assumptions 1.1 and 1.2 hold. The family of invariant probability mea- ε,δ 0 sures (μ )ε>0 converges weakly to the measure μ given by (1.5) in the sense that for every bounded and continuous function g : Rd × M → R,
T m0 ε,δ 1 lim g(x,i)μ (dx, i) = g(Xy(t))dt, ε→0 T i=1 Rd 0 ∈ = where T is the period of the limit cycle, y and g(x) i∈M g(x, i)νi .
ε Remark 1.4. Given that the switching component α is state-dependent with generator Q(x) = ∈ d = m0 (qij (x))M×M, x R , Theorem 1.1 still holds with f(x) i=1 f(x, i)νi(x) and (ν1(x), ..., = νm0 (x)) is the stationary distribution of a Markov chain with generator Q(x) (qij (x)) as long as Q(x) is bounded and satisfies the following conditions:
q (·) • For all i the functions q (·) and ij are Lipschitz continuous. ii qii(x·) • ∈ d qij (x) If qij (x) > 0for some x R then infx∈Rd |q (x)| > 0. • | | ii For all i we have infx∈Rd qii(x) > 0. • ˆ(m0) ˆ = ∨ ˆ(m0) infx∈Rd qij (x) > 0 where Q(x) (0 qij (x))M×M, and (qij (x)) is the m0-power of Q(x)ˆ
We explain how one can do this in Remark 2.1.
1.2. An application of Theorem 1.1
We will exhibit an example where the result of Theorem 1.1 applies. Recently there has been renewed interest in stochastic population dynamics [19,6,5,20]. Suppose we have a predator-prey system of the form ⎧ ⎪ d ⎨ x(t) = x(t)[a − bx(t) − y(t)h(x(t),y(t))] dt (1.8) ⎩⎪ d y(t) = y(t)[−c − dy(t) + x(t)f h(x(t), y(t))] . dt
Here x(t), y(t) denote the densities of prey and predator at time t ≥ 0, respectively; a, b, c, d, f> 0 describe the per-capita birth/death and competition rates, and xh(x, y), yh(x, y) are the func- tional responses of the predator and the prey. For instance, if h(x, y) is constant, the model is the classical Lotka-Volterra one [29,37,18]. If
m h(x,y) = 1 , m2 + m3x + m4y
320 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 the functional response is of Beddington-DeAngelis type [7]. The setting of (1.8)is very general and encompasses many of the models used in the ecological literature. We explore what happens in the fast-switching slow-noise limit for the following noisy exten- sion of (1.8) √ ε,δ = ε,δ ε,δ ε,δ ε + ε ε,δ dX (t) X (t)ϕ X (t), Y (t), α (t))dt √δλ(α (t))X (t)dW1(t) ε,δ ε,δ ε,δ ε,δ ε ε ε,δ (1.9) dY (t) = Y (t)ψ X (t), Y (t), α (t))dt + δρ(α (t))Y (t)dW2(t).
Here
ϕ(x,y,i) = a(i) − b(i)x − yh(x,y,i) and ψ(x,y,i) =−c(i) − d(i)y + f(i)xh(x,y,i), where a(·), b(·), c(·), d(·), f(·), λ(·), ρ(·) are positive functions defined on M, δ = δ(ε) depends ε on ε, lim δ = 0, W1(t) and W2(t) are independent Brownian motions, and α is an independent ε→0 Markov chain with generator Q/ε. As before, the generator Q is assumed to be irreducible so that the Markov chain has a unique stationary distribution given by (ν1, ..., νn0 ). The function 2 h(·, ·, ·) is assumed to be positive, bounded, and continuous on R+ × M. For g(·) = a(·), b(·), c(·), d(·), f(·), ϕ(·), ψ(·), define the averaged quantities g := g(i)νi , = { : ∈ M} = { : ∈ M} := gm min g(i) i , gM max g(i) i . Set h1(x, y) h(x, y, i)νi and h2(x, y) := f(i)h(x, y, i)νi . The existence and uniqueness of a global positive solution to (1.9) can be ε,δ = proved in the same manner as in [25]or [26] and is therefore omitted. We denote by Zz,i (t) ε,δ ε,δ ε = ∈ M ε,δ = ∈ 2 (Xz,i (t), Yz,i (t)) the solution to (1.9) with initial value α (0) i , Zz,i (0) z R+. Con- sider the averaged equation ⎧ ⎪ d ⎨ X(t) = X(t)ϕ(X(t), Y (t)) = X(t) a − bX(t) − Y(t)h1(X(t), Y (t)) dt (1.10) ⎩⎪ d Y(t)= Y(t)ψ(X(t), Y (t))) = Y(t) −c − dY(t)+ X(t)h (X(t), Y (t)) . dt 2
We denote by Zz(t) = (Xz(t), Y z(t)), the solution to (1.10) with initial value Zz(0) = z.
Assumption 1.3.
(i) The system (1.10) has a finite number of positive equilibria and a unique stable limit cycle . In addition, any positive solution not starting at an equilibrium converges to the stable limit cycle. (ii) The inequality a a h2 , 0 > c b b is satisfied.
Remark 1.5. Note that the Jacobian of xφ(x, y), yψ(x, y) at a , 0 has two eigenvalues: b 2 −c + a h ( a , 0) and − b < 0. If −c + a h ( a , 0) < 0, then a , 0 is a stable equilibrium of b 2 b a b 2 b b
321 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358
(1.10), which violates condition (i) of Assumption 1.3. This shows that condition (ii) is often contained in condition (i). We note that model (1.10)is quite general. Conditions on the parameters for the existence and uniqueness of a limit cycle are in general complicated.
We can apply Theorem 1.1 to this model if we can verify part (v) of Assumption 1.1 since the other conditions are clearly satisfied. Since the process αε(t) is ergodic and the diffusion is nondegenerate, an invariant probability measure of the solution Zε,δ(t) is unique if it exists. It is unlikely that one could find a Lyapunov-type function satisfying the hypothesis of [38, Theorem 3.26] in order to prove the existence of an invariant probability measure. In addition, ε,δ the tightness of the family of invariant probability measures (μ )ε>0 cannot be proved using the methods from [12,13]. These difficulties can be overcome with the help of a new technical tool. We partition the domain (0, ∞)2 into several parts and then construct a truncated Lyapunov-type function. We then estimate the average probability that the solution belongs to a specific part of our partition. ε,δ This then enables us to prove that the family of invariant probability measures (μ )ε>0 is tight 2 on the interior of R+, i.e., for any η>0, there are 0 <ε0, δ0 < 1 − με,δ([L 1,L]2) ≥ 1 − η. We are able to prove the following result. Theorem 1.2. Suppose Assumption 1.3 holds. For sufficiently small δ and ε, the process given ε,δ 2 2 by (1.9) has a unique invariant probability measure μ with support in Int R+ (where Int R+ 2 denotes the interior of R+). In addition: δ ε,δ a) If lim = l ∈ (0, ∞], the family of invariant probability measures (μ )ε>0 converges ε→0 ε weakly to μ0, the occupation measure of the limit cycle of (1.10), as ε → 0 (in the sense of Theorem 1.1). δ b) If lim = 0 and at each equilibrium (x∗, y∗) of (ϕ(x, y), ψ(x, y)), there is i∗ ∈ M such ε→0 ε that either ϕ(x∗, y∗, i∗) = 0 or ψ(x∗, y∗, i∗) = 0, then the family of invariant probability ε,δ 0 measures (μ )ε>0 converges weakly to μ , the occupation measure of the limit cycle of (1.10) as ε → 0. Remark 1.6. We note that on any finite time interval [0, T ], the solutions to (1.9) converge to the solutions of (1.10). However, in ecology, people are interested in the long-term behavior of ecosystems as T →∞. Therefore, the above result shows rigorously that (1.10)gives the correct long-term behavior. 1.3. Main ideas of proof of Theorem 1.1: a road map Because some parts of the proofs are very technical, in order to offer some insight, we present the main ideas in this subsection. It aims to provide something like a “road map” for the proofs. 322 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 Condition (v) of Assumption 1.1 is a tightness assumption for the family of invariant prob- ε,δ ε,δ ability measures (μ )0<ε<ε0 . This implies that any weak limit of (μ )0<ε<ε0 is an invariant probability measure of the limit system (1.4). The main technical issue is to show that any sub- ε,δ sequential limit of (μ )0<ε<ε0 does not assign any mass to any of the fixed points of f . This is done by a careful analysis of the nature of the deterministic and stochastic systems near the attracting region χl := {y : limt→∞ Xy(t) = xl}, of an equilibrium xl of f . Note that if xl is a source then χl ={xl} while if xl is hyperbolic χl can be an unbounded set. This makes the problem difficult. In Section 2, using large deviation techniques, we establish the following uniform estimate ε,δ for the probability that the processes Xx,i and Xx are close on a fixed time interval: For any R, T , and γ>0, there is a κ = κ(R, γ, T) > 0 such that κ P Xε,δ(t) − X (t) ≥ γ for some t ∈[0,T] < exp − ,x ∈ B . (1.11) x,i x ε + δ R The main task is to estimate the time of exiting the attracting region, χl ∩BR, of an equilibrium ε,δ ∩ xl. To be precise, we show that Xx,i leaves small neighborhoods of χl BR with strictly positive probability in finite time if we start close to χl ∩ BR. We find uniform lower bounds for these probabilities. In fact, for any sufficiently small > 0 and sufficiently large R>0, to include all the sets = Mi, i 1, ..., n0, we can find θ1, θ3 > 0, Hl > 0, and εl( ) such that for ε<εl( ), P τ ε,δ ≤ H ≥ ψ ,ε := exp − , |x − x | <θ , (1.12) x,i l ε + δ l 1 where ε,δ := { ≥ : ε,δ ∈ ε,δ ≥ } τx,i inf t 0 Xx,i(t) BR and dist(Xx,i(t), χl) θ3 . We prove the estimate (1.12)in the different cases as follows. ∗ ∗ 1) Suppose that there is an i ∈ M satisfying β f(xl, i ) = 0, where β is a normal unit vector ε ∗ of the stable manifold of (1.4)at xl. Then we estimate the time α (t) stays in i and consider the diffusion in this fixed state, that is √ ∗ ∗ dZδ(t) = f(Zδ(t), i )dt + δσ(Zδ(t), i )dW(t). Since the drift f(x, i∗) is nonzero and pushes us away from the stable manifold of x∗, and ε,δ the diffusion term is small, we can estimate the exit time τx,i . δ ∈ ∞] ∗ ∗ = δ ∞ 2) Suppose limε→0 ε (0, and there is an i such that β σ(xl, i ) 0. If limε→0 ε < , ε ∗ suppose in addition that β f(xl, i) = 0, i ∈ M. We estimate the time α (t) to be in i and consider the diffusion component in the direction β in this fixed state √ dZε,δ = δβ σ(Zε,δ,αε)dW(t) Since the diffusion coefficient does not vanish close to xl, we can do time change so that we get a Brownian motion. Then we can estimate the probability that the exit time exceeds a given number. Ultimately, we show that Zε,δ and β Xε,δ are close to each other. 323 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 Comparing the rates in (1.11) with (1.12)is key to prove the main result in Section 3 (see e.g. [22, 28]). The idea is to estimate the time of exiting the attracting region, χl ∩BR, of an equilibrium xl as well as the time of coming back to this region. Then we prove that eventually, the probability of entering χl ∩ BR is very small compared to the probability of exiting the region. If we start with X(0) close to χl ∩ BR then after a finite time X will be close to one of the equilibrium points or the limit cycle. Using this together with (1.11) and (1.12)we get that there exist neighborhoods N1, G1 of χl ∩ BR with N1 ⊂ G1 such that 1 P{τ ε,δ ε,δ = { ≥ : ε,δ ∈ \ } τx,i inf t 0 Xx,i(t) BR G1 . This can be leveraged into showing that with high probability, if we start in N1, we will leave the region G1 ⊃ N1 in a finite, uniformly bounded, time: 1 P τ ε,δ ε,δ →∞ → Note that T ,1 as ε 0. However, if we pick <κ/2, we have κ κ lim T ε,δ exp − = lim exp exp − = 0. (1.16) ε→0 ,1 ε + δ ε→0 ε + δ ε + δ The estimate (1.16)shows the exit time is not long compared to the good rate of large deviations, which will be used to show that invariant probability measures cannot put much mass on the equilibria. Let Xε,δ(t) be the stationary solution, whose distribution is με,δ for every time t ≥ 0. ε,δ ε,δ Let τ be the first exit time of X (t) from G1. We can show that for any η>0we can find ε,δ R>0 such that μ (N1) ≤ 2η by using (1.14), (1.15), and (1.16)to find the probabilities of the events ε,δ = ε,δ ε,δ ∈ ε,δ ≥ ε,δ ε,δ ∈ K1 X (T ,1) N1,τ T ,1, X (0) N1 ε,δ = ε,δ ε,δ ∈ ε,δ ε,δ ε,δ ∈ K2 X (T ,1) N1,τ 324 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 ε,δ = ε,δ ε,δ ∈ ε,δ ∈ \ K3 X (T ,1) N1, X (0) BR N1 ε,δ = ε,δ ε,δ ∈ ε,δ ∈ K4 X (T ,1) N1, X (0)/BR . Similar arguments show that for any η>0, we can find R>0 and neighborhoods ∩ ∩ N1, ..., Nn0−1 of χ1 BR, ..., χn0−1 BR such that − ε,δ ∪n0 1 ≤ n0 lim sup μ ( j=1 Nj ) 2 η. ε→0 Using this fact together with Assumption 1.1 and Lemma 2.2 we can establish, by a straightfor- ward modification of the proof of [22, Theorem 1], that for any η>0 there is neighborhood N of the limit cycle such that lim inf με,δ(N) > 1 − 2n0 η. ε→0 2. Estimates of the first exit times Define for any i = 1, ..., n0 and θ>0, the sets χi := {y : limt→∞ dist(Xy(t), Mi) = 0} and := { : } Mi,θ y dist(y, Mi) <θ . Let R0 > 1be large enough such that BR0−1 contains all Mi , = ∈ { = } ∩ i 1, ..., n0. Fix θ0 (0, 1) such that Mi,2θ0 , i 1, ..., n0 are mutually disjoint and Mi,2θ0 ε,δ χj =∅for j F Lemma 2.1. (Exponential martingale inequality) Suppose (g(t)) is a real-valued t -adapted T 2 ∞ process and 0 g (t)dt < almost surely. Then for any a, b>0 one has ⎧ ⎡ ⎤ ⎫ ⎨ t t ⎬ ⎣ − a 2 ⎦ ≤ −ab P ⎩ sup g(s)dW(s) g (s)ds >b⎭ e . t∈[0,T ] 2 0 0 We will make use of this lemma repeatedly in the proofs to follow. The next result gives us estimates on how close the solutions to (1.3) and (1.7)are on a finite time interval if they have the same starting points. The argument of the proof is pretty standard. For completeness, it relegated to Appendix A. Lemma 2.2. For any R, T , and γ>0, there is a κ = κ(R, γ, T) > 0 such that κ P Xε,δ(t) − X (t) ≥ γ for some t ∈[0,T] < exp − ,x∈ B . x,i x ε + δ R d ˇε,δ Lemma 2.3. Let N be an open set in R and let τx,i be any stopping time. Suppose that there is ∈ × M {ˇε,δ } ≥ ε,δ an > 0 such that for all starting points (x, i) N one has P τx,i < a > 0, where lim aε,δ = 0. Then P τˇε,δ < > 1/2 for (x, i) ∈ N × M if ε is sufficiently small. ε→0 x,i aε,δ 325 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 Lemma 2.4. The following properties hold: (1) For any θ>0, R>0, there exists T1 > 0 such that for any y ∈ BR, Xy(t) ∈ Mk,θ for some ∈{ } t The following lemmas show that the process leaves small neighborhoods around the equi- librium points with strictly positive probability in finite time if we start close to the equilibrium points. Furthermore, this probability can be bounded below uniformly for all starting points close to the equilibrium. We need this because we want to show the convergence of the process to the limit cycle . ∗ ∗ Lemma 2.5. Consider an equilibrium xl and suppose there exists i ∈ M such that β f(xl, i ) = 0 where β is a normal unit vector of the stable manifold of (1.4) at xl. Then for any > 0 that is sufficiently small and any R>R0, we can find θ1, θ3 > 0, Hl > 0, and εl( ) such that for ε<εl( ), P τ ε,δ ≤ H ≥ ψ ,ε := exp − ,x∈ M , x,i l ε l,θ1 where ε,δ := { ≥ : ε,δ ∈ ε,δ ≥ } τx,i inf t 0 Xx,i(t) BR and dist(Xx,i(t), χl) θ3 . Proof. Suppose without loss of generality that xl = 0. Let β be a normal vector of the stable manifold at 0 such that |β| = 1 and β f(0, i∗) > 0. Since f is locally Lipschitz we can find a1 > 0 such that ∗ β f(x,i )>a1 > 0, |x| <θ0. (2.1) |f(x,i∗)| A := ∗ < ∞ Then 1 supx<θ0 β f(x,i ) . Since β is perpendicular to the tangent of the stable manifold at 0, we can find θ2 ∈ 1 a1 0, ∧ θ0 such that 2+3A1 4|qi∗i∗ | θ2 := dist(Ll ,χl) θ3 > 0, (2.2) where θ2 ={ :| |≤ + | | } Ll x x (2 3A1)θ2 and β x >θ2 . (2.3) The continuous dependence of the solutions of (1.4)on the starting point and the fact that 0is an equilibrium of (1.4)imply that X stays close to 0for a finite time if the starting point is close 326 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 enough to 0. Using this, we can derive from Lemma 2.2 that there exist numbers θ1 ∈ (0, θ2) and k>0 such that | ε,δ | + 1 − − k ∈ ∈ M P Xx,i(t) <θ2, 0 ∗ First, we consider" the case #αε(0) = i . Because of the independence of αε(·) and W(·), if ε = ∗ ∈ ε,δ · α (t) i for all t 0, | ∗ ∗ | , the process X ∗ ( ) has the same distribution on the time inter- " # qi i x,i val 0, as that of Zδ given by |qi∗i∗ | x √ ∗ ∗ dZδ(t) = f(Zδ(t), i )dt + δσ(Zδ(t), i )dW(t). (2.5) Define the bounded stopping time ε,δ := ∧ { :| δ |≥ }∧ { : δ ≥ } ρx inf t>0 Zx(t) θ0 inf t>0 β Zx(t) θ2 . |qi∗i∗ | We have ε,δ ε,δ ρx ρx √ δ ε,δ = + δ ∗ + δ ∗ | |≤ β Zx(ρx ) β x β f(Zx(s), i )ds δβ σ(Zx(s), i )dW(s), x θ0. (2.6) 0 0 By the exponential martingale inequality from Lemma 2.1, there exists a constant m3 > 0 inde- pendent of δ such that 3 3 P ε,δ,1 > and P ε,δ,2 > x 4 x,i 4 where t √ ε,δ,1 := − δ ∗ x δβ σ(Zx(s), i )dW(s) 0 t $ √ 1 δ ∗ δ ∗ ε,δ −√ δβ σ(Z (s), i )σ (Z (s), i ) βds < m3 δ,t ∈ 0,ρ δ x x x 0 and t √ ε,δ, δ ∗ 2 := δσ(Z (s), i )dW(s) x x 0 t $ √ −√1 δ ∗ δ ∗ ∈ ε,δ δ σ(Zx(s), i )σ (Zx(s), i ) ds < m3 δ,t 0,ρx . δ 0 327 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 This implies that 1 P ε,δ,1 ∩ ε,δ,2 > . (2.7) x x,i 2 ε,δ,1 Using (2.1) and (2.6), we note that on the set x ε,δ ρx δ ε,δ + δ ∗ β Zx(ρx )>βx β f(Zx(s), i )ds 0 ε,δ ρx √ 1 δ ∗ δ ∗ − √ β δσ(Z (s), i ) σ(Z (s), i )βds − m3 δ (2.8) δ x x 0 ε,δ ρx √ ≥−θ2 + a1ds − m3 δ. 0 √ a1 ε,δ = ∈ ε,δ,1 ≤ Let δ be so small that m3 δ< . If ρx (ω) for some ω x,i , using θ2 2 |qi∗i∗ | |qi∗i∗ | a a ρε,δ 1 = 1 x , we get 4|qi∗i∗ | 4 √ | δ ε,δ |≤ − + ε,δ − β Zx(ρx (ω)) θ2 < θ2 a1ρx m3 δ, ε,δ,1 which contradicts (2.8). As a result, if x ≤ θ2, ω ∈ x and δ is sufficiently small, we have ε,δ ρx (ω) < , (2.9) |qi∗i∗ | and by (2.6)we have ε,δ ε,δ ρx ρx √ √ δ ∗ ≤| δ ε,δ |+| |+ δ ∗ δ ∗ + β f(Zx(s), i )ds β Zx(ρx ) β x δ σ(Zx(s), i )σ (Zx(s), i ) ds m3 δ 0 0 <3θ2 (2.10) ε,δ,1 ε,δ,2 on x ∩ x . Using (2.5) and (2.10), one sees that if δ is sufficiently small and |x| <θ2, ε,δ,1 ε,δ,2 then for ω ∈ x ∩ x , ε,δ ε,δ ρx ρx √ √ | ε,δ | | |+ | ∗ | + δ ∗ δ ∗ + Zx(ρx ) < x f(Zx(t), i ) dt δ σ(Zx(s), i )σ (Zx(s), i ) ds m3 δ 0 0 328 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 ε,δ ρx ∗ <2θ2 + A1 β f(Zx(t), i )dt 0 <(2 + 3A1)θ2 <θ0. (2.11) ε,δ ε,δ ε,δ Combining (2.11) with the definition of ρx shows that β Zx(ρx ) = θ2 and |Zx(ρx )| <(2 + ε,δ,1 ε,δ,2 3A1)θ2 on x ∩ x . As a result of this and (2.7), if |x| ≤ θ2, % & ≥ | |≤ + ∈ ≥ ε,δ,1 ∩ ε,δ,2 1 P β Zx(t) θ2, Zx(t) (2 3A1)θ2 for some t 0, P x x,i > . |qi∗i∗ | 2 Let ε,δ := { : ε,δ ≥ | ε,δ|≤ + }= { : ε,δ ∈ θ2 } ζx,i inf t>0 β Xx,i(t) θ2, Xx,i (2 3A1)θ2 inf t>0 Xx,i Ll . Using the independence of αε, the paragraph before equation (2.5), and the last two equations, we obtain % & ε,δ ≤ 1 ε = ∗ ∈ = 1 − | |≤ P ζx,i∗ > P αi∗ (t) i , for all t 0, exp , if x θ1. |qi∗i∗ | 2 |qi∗i∗ | 2 ε (2.12) Since αε(t) is ergodic, for any sufficiently small ε, i.e., small enough , ∗ 3 P{αε(t) = i for some t ∈[0, 1]} > ,i∈ M. (2.13) i 4 By the strong Markov property, we derive from (2.4), (2.12), and (2.13) that for all (x, i) ∈ × M Ml,θ1 and for ε sufficiently small ε,δ + ≥ 1 − P ζx,i < 1 exp . (2.14) |qi∗i∗ | 4 ε The proof is complete by combining this estimate with (2.2). δ Lemma 2.6. Suppose that lim = r>0. Assume that at the equilibrium point xl, one has ε→0 ε ∗ ∗ f(xl, i) = 0 for all i ∈ M, and there is i ∈ M for which β σ(xl, i ) = 0, where β is a nor- mal unit vector of the stable manifold of (1.4) at xl. Then for any sufficiently small > 0 and any R>R0, we can find θ1, θ3 > 0, Hl > 0, and εl( ) > 0 such that for ε<εl( ), P τ ε,δ ≤ H ≥ ψ ,ε := exp − , for all (x, i) ∈ M × M, x,i 1 δ l,θ1 where ε,δ = { ≥ : ε,δ ∈ ε,δ ≥ } τx,i inf t 0 Xx,i(t) BR and dist(Xx,i(t), χl) θ3 . 329 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 δ Proof. We can assume without loss of generality that xl = 0 and lim = 1. Since σ is locally ε→0 ε Lipschitz, we can find a2 > 0 such that ∗ a2 <β(σ σ )(y, i )β, |y| <θ0. (2.15) Let Kl > 0be such that |f(x, i)| 2 Kl T (2 + KlT) e θ1 <θ0 (2.16) θ1 := and dist(Ll , χl) θ3 > 0 where θ1 := { :| |≤ + 2 Kl T | | } Ll x x (2 KlT) e θ1 and β x >θ1 . (2.17) Define ⎧ '' ( ( ⎫ ⎨ u ⎬ := : ∧ θ0 ε,δ ε ≥ ζt,x,i inf u>0 β (σ σ ) 1 X (s), αi (s) βds t . ⎩ |Xε,δ(s)| x,i ⎭ 0 x,i ≥ ε For all t 0, we have by (2.15) and the ergodicity of the Markov chain αi that P(ζt,x,i < ∞) = 1, |x| <θ0. As a result the process (M(t))t≥0 defined by '' ( ( ζ t,x,i = ∧ θ0 ε,δ ε M(t) β σ 1 X (s), αi (s) dW(s) |Xε,δ(s)| x,i 0 x,i is a Brownian motion. This follows from the fact that M(t) is a continuous martingale with quadratic variation [M, M]t = t, t ≥ 0. Set θ2 := (2 + KlT)θ1. Since M(1) has the distribution of a standard normal, for sufficiently small δ, we have the estimate ' ( √ 1 θ 2 P{ δM(1)>θ }≥ exp − 2 , |x| <θ . (2.18) 2 2 δ 0 Using the large deviation principle (see [24]), we can find a3 = a3(T ) > 0 such that ⎧ ⎫ ⎨ T ⎬ 1 νi∗ a3 P 1{αε(s)=i∗}ds > ≥ 1 − exp − . (2.19) ⎩T i 2 ⎭ ε 0 330 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 a ν ∗ T Equation (2.15), the definition of ζ , and 2 i > 1 yield t,x,i 2 ⎧ '' ( ( ⎫ ⎨ T ⎬ ∗ ∧ θ0 ε,δ ε ≥ a2νi T ≥ − −a3 P β (σ σ ) 1 X (s), αi (s) βds 1 exp , ⎩ |Xε,δ(s)| x,i 2 ⎭ ε 0 x,i |x| <θ0, which leads to a P{ζ ≤ T }≥1 − exp − 3 , |x| <θ . (2.20) 1,x,i ε 0 Define for |x| <θ0, i ∈ M t '' ( ( √ ε,δ,3 := ∧ θ0 ε,δ ε δ σ 1 X (s), αi (s) dW(s) x,i |Xε,δ(s)| x,i 0 x,i '' ( ( t θ θ < 2 δ (σ σ) 1 ∧ 0 Xε,δ(s), αε(s) ds + θ δ | ε,δ | x,i i 2 Xx,i(s) 0 ≤ (KlT + 1)θ2,t ∈[0,T] and note that the last inequality holds by the definition of Kl. By Lemma 2.1 ' ( 2θ 2 P( ε,δ,3) ≥ 1 − exp − 2 , |x| <θ . (2.21) x,i δ 0 Define the stopping time = { :| ε,δ |≥ }∧ { :| ε,δ |≥ + Kl T } ζx,i inf t>0 β Xx,i(t) θ1 inf t>0 Xx,i(t) (Kl 2)θ2e . √ | | ≤ ∈{ } ∩{ ≤ } ∩ ε,δ,3 If x θ1 and ω δM(1) >θ2 ζ1,x,i T x,i , we claim that we must have ζx,i ζ '' ( ( √ √ 1,x,i = + = ∧ θ0 ε,δ ε θ2 (2 KlT)θ1 < δM(1) δ β σ 1 X (s), αi (s) dW(s) |Xε,δ(s)| x,i 0 x,i 331 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 ζ 1,x,i ≤| ε,δ |+| |+ ε,δ ε β Xx,i(ζ1) β x β f(Xx,i(s), αi (s))ds 0 ζ 1,x,i ≤ + | ε,δ | + = 2θ1 Kl β Xx,i(s) ds < (2 KlT)θ1 θ2, 0 ∧ θ0 ε,δ = ε,δ where we used that 1 | ε,δ | Xx,i(s) Xx,i(s) if s<ζx,i by the definition of ζx,i and Xx,i(s) (2.16). √ | | ≤ ∈{ } ∩{ ≤ } ∩ ε,δ,3 ≤ ≤ ≤ For x θ1 and ω δM(1) >θ2 ζx,i T x,i , for any 0 t ζ1,x,i T , t t √ ε,δ ε,δ ε ε,δ ε |X (t)|≤|x|+ δ σ X (s), α (s) dW(s) + |f(X (s), α (s))|ds x,i x,i i x,i i 0 0 t + + | ε,δ | <(KlT 2)θ2 Kl Xx,i(s) ds. 0 This together with Gronwall’s inequality implies that | ε,δ | + Kl T ∈[ ] Xx,i(t) <(KlT 2)θ2e ,t 0,ζx,i √ | | ≤ ∈{ } ∩{ ≤ } ∩ ε,δ,3 Thus for x θ1 and ω δM(1) >θ2 ζx,i T x,i , we have that ζx,i ε,δ + Kl T ε,δ ≥ Xx,i(ζx,i) <(KlT 2)θ2e and β Xx,i(ζx,i) θ1. δ = Since θ2 2 if <θ2 , which completes the proof. δ Lemma 2.7. Suppose that lim =∞. Assume that at the equilibrium point xl one can find ε→0 ε ∗ ∗ i ∈ M such that β σ(xl, i ) = 0 where β is a normal unit vector of the stable manifold of (1.4) at xl. Then for any sufficiently small > 0 and any R P τ ε,δ ≤ H ≥ ψ ,ε := exp − for all (x, i) ∈ M × M, x,i l δ l,θ1 where ε,δ = { ≥ : ε,δ ∈ ε,δ ≥ } τx,i inf t 0 Xx,i(t) BR and dist(Xx,i(t), χl) θ3 . 332 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 Proof. Assume, as in the previous lemmas, that xl = 0. Pick a number a2 > 0for which ∗ a2 <β(σ σ )(y, i )β, |y| <θ0. Let Kl > 0be such that |f(x)| θ1 ={ :| − |≤ + 2 Kl T | − | } Ll x x xl (3 KlT) e θ1 and β (x xl) >θ1 . (2.23) Define θ2 = (3 + KlT)θ1 and let a2, M(t), T, ζ1,x,i be as in the proof of Lemma 2.6. Arguing as in the proof of (2.20), we can find a3 > 0 such that ) * a P ζ ≤ T ≥ 1 − exp − 3 , |x| <θ . 1,x,i ε 0 Since f(0) = 0, we can apply the large deviation principle (see [24]) to show that there is κ = κ( ) > 0 such that κ P(A) ≥ 1 − exp − , (2.24) ε ) * := u ε ∈[ ] where A 0 f(0,αi (s))ds <θ1, for all u 0,T . The estimates ζ 1,x,i = ε,δ ε M(1) β σ(Xx,i(s), αi (s))dW(s) 0 ζ 1,x,i ≤| ε,δ |+| |+ ε β Xx,i(ζ1,x,i ) β x β f(0,αi (s))ds 0 ζ 1,x,i + ε,δ ε − ε β f(Xx,i(s), αi (s)) f(0,αi (s)) ds, 0 and t t √ ε,δ ε,δ ε ε,δ |X (t)|≤|x|+ δ σ X (s), α (s) dW(s) + |f(X (s))|ds x,i x,i i x,i 0 0 t + | ε,δ − ε,δ ε | f(Xx,i(s)) f(Xx,i(s), αi (s) ds 0 together with arguments similar to those from the proof of Lemma 2.6 show that 333 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 1 P Xε,δ(t) ∈ Lθ1 for some t ∈[0,T] ≥ exp − ,(x,i)∈ M × M x,i l 4 δ l,θ1 if δ is sufficiently small. Remark 2.1. The results in this section still hold true if one assumes the generator Q(·) of α(·) is state dependent; see an explanation of the exact setting in Remark 1.4. By the large deviation principle in [3, Section 3] and the truncation arguments in Lemma A.1, we can obtain Lemma 2.2 for the case of state-dependent switching. It should be noted that while [3] only considers Case 1 of (1.6), using the variational representation, the arguments in [3, Section 3] can be applied to obtain Lemma 2.2 for the other cases. We can also infer from the large deviation principle that (2.13), (2.19) and (2.24) hold in this setting. As a result, Lemmas 2.5, 2.6 and 2.7 hold. These lemmas, in combination with the proofs from Section 3 imply that the main result, Theorem 1.1, remains unchanged if one has state-dependent switching. 3. Proof of main result This section provides the proofs of the convergence of με,δ for the three cases given in (1.6). Proposition 3.1. For every η>0, there exists R>R0 and neighborhoods N1, ..., Nn0−1 of ∩ ∩ χ1 BR, ..., χn0−1 BR such that − ε,δ ∪n0 1 ≤ n0 lim sup μ ( j=1 Nj ) 2 η. ε→0 ε,δ Proof. For any η>0, let R>R0 be such that μ (BR) ≥ 1 − η. Define S1 ={y ∈ BR : dist(y, χ1 ∩ BR)<θ0}. In view of Lemma 2.4, there exists c2 > 0 such that for all t ≥ 0 dist(Xy(t), χ1) ≥ 2c2 for any y ∈ BR \ S1. (3.1) Define G1 ={y ∈ BR : dist(y, χ1 ∩ BR) There exists c3 > 0 such that dist(Xy(t), χ1) ≥ 2c3 for any y ∈ BR \ G1,t≥ 0. (3.2) Note that we have 2c3 ≤ c2 and 2c2 ≤ θ0. Define N1 ={y ∈ BR : dist(y, χ1 ∩ BR) 334 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 ∈ ∈ ∩ \ In view of Lemma 2.4, for any y/χ1, there exists ty such that Xy(ty) Mi,θ0 (BR S1) for some i>1. This together with the continuous dependence of solutions to initial values and (3.1) implies that there exists T>ˆ 0 such that ˆ dist(Xy(t), χ1) ≥ 2c2 for any t ≥ T,y∈ BR \ N1. (3.3) = ˆ κ Let κ κ(R, c3, T)be as in Lemma 2.2, < 2 , θ1, and ψε be as in one of the Lemmas 2.5, 2.6 and 2.7 (depending on which case we are considering). We have ε,δ ≥ ∈ P(τx,i ε,δ = { ≥ : ε,δ ∈ ε,δ ≥ } τx,i inf t 0 Xx,i(t) BR and dist(Xx,i(t), χ1) θ3 . Define ε,δ = { ≥ : ε,δ ∈ \ } τx,i inf t 0 Xx,i(t) BR G1 . ∈ It follows! from part (1) of Lemma 2.4 that for any x N1, there exists a T1 > 0 such that n0 X (t ) ∈ M θ for some t ≤ T . x x j=1 j, 1 x 1 2 ! ! ∈ n0 n0 ∩ =∅ Suppose Xx(tx) j=2 M θ1 . Note that j=2 M θ1 M1,c3 , θ1 <θ0 and that by con- j, 2 j, 2 ! ∩ =∅ n0 ∩ =∅ struction, M1,2θ0 χj , j>1. These facts imply that j=2 M θ1 N1 . This together j, 2 with Lemma 2.2 and (3.3) implies 1 P{τ ε,δ < T˜ + Tˆ } > (3.5) x,i 1 2 for small ε>0. = θ1 ∈ When ε is sufficiently small, we have by Lemma 2.2 (applied with γ 2 ) that for any x N1 ∈ satisfying Xx(tx) M θ1 that 1, 2 1 P{Xε,δ(t ) ∈ M } > . (3.6) x,i x 1,θ1 2 Similarly to (3.3), there exists a T2 > 0 such that dist(Xy(t), χ1) ≥ 2c2 for any t ≥ T2,y∈ BR, dist(y, χ1) ≥ θ3, which implies that by Lemma 2.2, for sufficiently small ε>0, 1 P dist(Xε,δ(T ), χ ) ≥ c > for any x ∈ B , dist(x, χ ) ≥ θ ,i∈ M. (3.7) x,i 2 1 2 2 R 1 3 335 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 Putting together (3.4), (3.6), and (3.7)we deduce that 1 P{τ ε,δ < T + H + T } > ψ , (3.8) x,i 1 2 4 ε for ε sufficiently small. Combining (3.5) and (3.8), we get that 1 P{τ ε,δ + + + ˆ ε,δ := H T1 T2 T Define T ,1 4 . Applying Lemma 2.3 to (3.9), we have ψε,δ 1 P τ ε,δ ε,δ ε,δ We will argue by contradiction that lim supμ (N1) ≤ 2η. Assume that lim sup μ (N1) > 2η> ε→0 ε→0 0. Since <κ/2, we have κ lim T ε,δ exp − = 0. (3.11) ε→0 ,1 ε + δ Let Xε,δ(t) be the stationary solution, whose distribution is με,δ for every time t ≥ 0. Let τ ε,δ be ε,δ the first exit time of X (t) from G1. Define the events ε,δ = ε,δ ε,δ ∈ ε,δ ≥ ε,δ ε,δ ∈ K1 X (T ,1) N1,τ T ,1, X (0) N1 ε,δ = ε,δ ε,δ ∈ ε,δ ε,δ ε,δ ∈ K2 X (T ,1) N1,τ Note that the above events are disjoint with union N1 such that 4 ε,δ = { ε,δ} μ (N1) P Kn . n=1 Using (3.10), we get that 1 P(Kε,δ) ≤ με,δ(N ) and P(Kε,δ) ≤ 1 − με,δ(B )<η. (3.12) 1 2 1 4 R ε,δ Next, we estimate P(K3 ). It follows from Lemma 2.2, (3.2), and (3.3) that if ε is sufficiently small then 336 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 κ P Xε,δ(T)/ˆ ∈ G ≥ 1 − exp − ,x∈ B \ N , x,i 1 ε + δ R 1 and κ P Xε,δ(t)∈ / N , for all t ∈[0, Tˆ ] ≥ 1 − exp − ,x∈ B \ G . x,i 1 ε + δ R 1 Using the last two estimates together with the Markov property one sees that for any x ∈ BR \ ∈ M ∈[ ε,δ ] G1, i , s 0, T ,1 , ε,δ ∈ P Xx,i(s) N1 = ε,δ ∈ ε,δ ˆ ∈ \ P Xx,i(s) N1,Xx,i(T)/BR G1 s/Tˆ + ε,δ ∈ ε,δ ˆ ∈ \ ε,δ ˆ ∈ \ = − P Xx,i(s) N1,Xx,i(nT)/BR G1,Xx,i(ιT) BR G1,ι 1, ..., n 1 = n2 + ε,δ ∈ ε,δ ˆ ∈ \ = [ ˆ ] P Xx,i(s) N1,Xx,i(ιT) BR G1,ι 1, ..., s/T s/Tˆ ≤ ε,δ ˆ ∈ \ + ε,δ ˆ ∈ \ ε,δ − ˆ ∈ \ } P Xx,i(T)/BR G1 P Xx,i(nT)/BR G1,Xx,i((n 1)T) BR G1 = n 2 "+ , + , # + , + ε,δ ∈ ∈ ˆ ˆ ˆ ˆ + ˆ ε,δ ˆ ˆ ∈ \ P Xx,i(t) N1, for some t s/T T, s/T T T ,Xx,i s/T T BR G1 + , κ ≤ s/Tˆ + 1 exp − ε + δ κ ≤ s/Tˆ + 1 exp − , ε + δ (3.13) where s/Tˆ denotes the integer part of s/Tˆ . ∈[ˆ ε,δ] ∈ \ Note that similar arguments show that (3.13)also holds for all s T, Tx,i and x BR N1. = ε,δ It follows from this with s T ,1, κ P(Kε,δ) = P Xε,δ(T ε,δ ) ∈ N , Xε,δ(0) ∈ B \ N ≤ T ε,δ /Tˆ + 1 exp − . 3 ,1 1 R 1 ,1 ε + δ This together with (3.11) implies that lim P(Kε,δ) = 0. (3.14) ε→0 3 Using (3.13) and the strong Markov property, we get ε,δ = ε,δ ε,δ ∈ ε,δ ε,δ ε,δ ∈ P(K2 ) P X (T ,1) N1,τ 337 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 ⎡ ⎤ T ε,δ ,1 ⎢ ) *⎥ = { ε,δ ∈ } ⎣ ε,δ ε,δ − ∈ ε = ε,δ ∈ ⎦ P τ dt P Xx,i(T ,1 t) N1 P α (t) i, X (t) dx i∈M 0 ∂G1 κ ≤ T ε,δ /Tˆ + 1 exp − ,1 ε + δ →0asε → 0 due to (3.11). (3.15) Putting together the estimates (3.12), (3.15), and (3.14), we see that ε,δ 1 ε,δ lim sup μ (N1) ≤ lim sup μ (N1) + 0 + 0 + η, ε→0 2 ε→0 ε,δ which contradicts the assumption that lim supμ (N1) > 2η. Thus, we have shown that ε→0 ε,δ lim μ (N1) ≤ 2μ. ε→0 Define S2 ={y ∈ BR \ S1 : dist(y, χ2 ∩ BR \ S1)<θ0}. There exists c4 > 0 such that dist(Xy(t), χ1) ≥ 2c4 for any y ∈ BR \ S1. Define G2 ={y ∈ BR \ S1 : dist(y, χ2 ∩ (BR \ S1)) < c4} There exists c5 > 0 such that dist(Xy(t), χ1) ≥ 2c5 for any y ∈ BR \ G1. Define N2 ={y ∈ BR \ S1 : dist(y, χ1 ∩ BR \ S1) ˆ ˆ ∈ \ ∪ ∈ \ ∪ Let T2 be such that Xy(T2) BR (S1 S2) given that y BR (S1 N2). We can show, just as ε,δ ε,δ κ above, that there exists a T such that lim → T exp − = 0 and ,2 ε 0 ,2 ε + δ 1 P{τ ε,δ 338 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 ε,δ ≤ Applying the same arguments as in the previous part, we can show that lim supε→0 μ (N2) ∩ 4η. Continuing this process, we can construct neighborhoods N1, ..., Nn0−1 of χ1 BR, ..., ∩ χn0−1 BR such that − ε,δ ∪n0 1 ≤ n0 lim sup μ ( j=1 Nj ) 2 η. ε→0 Theorem 1.1. Suppose Assumptions 1.1 and 1.2 hold. The family of invariant probability mea- ε,δ 0 sures (μ )ε>0 converges weakly to the measure μ given by (1.5) in the sense that for every bounded and continuous function g : Rd × M → R, T m0 ε,δ 1 lim g(x,i)μ (dx, i) = g(Xy(t))dt, ε→0 T i=1 Rd 0 ∈ = where T is the period of the limit cycle, y and g(x) i∈M g(x, i)νi . Proof. We have proved in Proposition 3.1 that for any η>0we can find R>0 and neighbor- ∩ ∩ hoods N1, ..., Nn0−1 of χ1 BR, ..., χn0−1 BR such that + ε,δ ∪n0 ≤ n0 1 lim sup μ ( j=1Nj ) 2 η. ε→0 Using this fact together with Assumption 1.1 and Lemma 2.2, by a straightforward modification of the proof of [22, Theorem 1], we can establish that for any ϑ>0 there is neighborhood N of the limit cycle such that lim inf με,δ(N) > 1 − ϑ. ε→0 4. Proof of Theorem 1.2 To proceed, we first need some auxiliary results. Lemma 4.1. There exist numbers K1 and K2 > 0 such that for any 0 <ε, δ<1 and any (i0, z0) ∈ 2 M × Int R+, we have t 1 ε,δ 2 E |Z (s)| ds ≤ K1(1 +|z0|), t ≥ 1, t z0,i0 0 and | ε,δ |2 ≤ lim sup E Zz ,i (t) K2. t→∞ 0 0 339 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 Proof. Let θ ˆ 2 2 K1 = sup {fM x(a(i) − b(i)x) − y(c(i) + d(i)y)+ θ(x + y )} < ∞. 2 (x,y,i)∈R+×M ˆ ε,δ ˆ ˆ 2 2 ε,δ Consider V(x, y, i) = fM x +y. We can check that L V(x, y, i) ≤ K1 −θ(x +y ), where L the generator associated with (1.9)(see [32, p. 48] or [38]for the formula of Lε,δ). Similarly, we ˆ ε,δ ˆ 2 ˆ ˆ 2 can verify that there is K2 > 0 such that for all ε<1, δ<1, L (V (x, y, i)) ≤ K2 −V (x, y, i). For each k>0, define the stopping time σk = inf{t : x(t) + y(t) >k}. By the generalized Itô formula for V(x(t),ˆ y(t), αε(t)) t ∧σk EV(Zˆ ε,δ (t ∧ σ ), αε(t ∧ σ )) = V(zˆ ,i ) + E Lε,δV(Zˆ ε,δ (s), αε(s))ds z0,i0 k k 0 0 z0,i0 0 (4.1) t ∧σ k ≤ f x + y + E Kˆ − θ|Zε,δ (s)|2 ds. M 0 0 1 z0,i0 0 Hence t ∧σk θE |Zε,δ (s)|2ds ≤ f x + y + Kˆ t. z0,i0 M 0 0 1 0 Letting k →∞and dividing both sides by θt we have t 1 f x + y Kˆ E |Zε,δ (s)|2ds ≤ M 0 0 + 1 . (4.2) t z0,i0 θt θ 0 Applying the generalized Itô formula to et Vˆ 2(Zε,δ (t), αε(t)), z0,i0 ∧ Eet σk Vˆ 2(Zε,δ (t ∧ σ ), αε(t ∧ σ )) z0,i0 k k t ∧σk = Vˆ 2(z ,i ) + E es (Vˆ 2(Zε,δ (s), αε(s)) + Lε,δVˆ 2(Zε,δ (s), αε(s)) ds 0 0 z0,i0 z0,i0 0 (4.3) t ∧σk 2 ˆ s 2 ˆ t ≤ (fM x0 + y0) + K2E e ds ≤ (fM x0 + y0) + K2e . 0 Taking the limit as k →∞, and then dividing both sides by et , we have − E f Xε,δ (t) + Y ε,δ (t) 2 ≤ (f x + y )2e t + Kˆ . (4.4) M z0,i0 z0,i0 M 0 0 2 The assertions of the lemma follow directly from (4.2) and (4.4). 340 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 Lemma 4.2. There is a number K3 > 0 such that t " # 1 E ϕ2(Zε,δ(s), αε(s)) + ψ2(Zε,δ(s), αε(s)) ds ≤ K (1 +|z|) t z,i i z,i i 3 0 2 for all ε, δ ∈ (0, 1], z ∈ Int R+, t ≥ 1. Proof. Since the function h(·, ·, i) is bounded, we can find C>0 such that ϕ2(z, i) + ψ2(z, i) ≤ C(1 +|z|2). The claim follows by an application of Lemma 4.1. Recall that the two equilibria of (1.10)on the boundary are both hyperbolic. Note that the Jacobian of xφ(x, y), yψ(x, y) at a , 0 has two eigenvalues: −c + a h a , 0 > 0 and b b 2 b 2 − b − a < 0. At (0, 0), the two eigenvalues are a>0 and c<0, respectively. If we consider the 2c weighted average Lyapunov exponent, we can see that the growth rate of d ln X(t) + d ln Y(t) a dt dt 2c is positive both at (0, 0) and a , 0 . This suggests we should look at d ln X(t) + d ln Y(t) in b a dt dt order to prove that the dynamics of (1.10)is pushed away from the boundary. Then we can use ε,δ 2 approximation arguments to obtain the tightness of (Z ) on Int R+. Define 2c ϒ(z,i) := ϕ(z,i) + ψ(z,i) a and 2c ϒ(z) := ϕ(z) + ψ(z). a We have the following lemma. 1 Lemma 4.3. Let γ = c ∧ − c + a h ( a , 0) > 0. For any H>a + 1, there are numbers 0 2 b 1 b b 2 T, β>0 such that for all z ∈{(x, y) ∈ R+ | x ∧ y ≤ β, x ∨ y ≤ H } T 1 X (T ) ∨ Y (T ) ≤ H and ϒ(Z (t))dt ≥ γ . (4.5) z z T z 0 0 Proof. Since lim Z(0,y)(t) = (0, 0), ∀y ∈ R+ and t→∞ 2c 2c ϒ(0, 0) = ϕ(0, 0) + ψ(0, 0) = a − c = c ≥ 2γ , (4.6) a a 0 341 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 there exists T1 > 0 such that t 1 3 ϒ(Z (s))ds ≥ γ for t ≥ T ,y∈[0,H]. (4.7) t (0,y) 2 0 1 0 By (4.6) and the continuity of ϒ(·), there exists β ∈ (0, a ) such that 1 b 7 ϒ(x,0) ≥ γ , if x ≤ β . (4.8) 4 0 1 Since a 2c a a a a ϒ , 0 = ϕ , 0 + ψ , 0 =−c + h1 , 0 ≥ 2γ0 b a b b b b and a lim Z(x,0)(t) → , 0 , ∀x>0, t→∞ b there exists a T2 > 0 such that t 1 7 ϒ(Z (s))ds ≥ γ for t ≥ T ,x∈[β ,H]. (4.9) t (x,0) 4 0 2 1 0 ) * MH Let MH = sup ∈[ ] |ϒ(x,0)| , tx = inf{t ≥ 0 : Xx, ≥ β } and T = 4 + 7 T . It can x 0,H 0 1 3 γ0 2 ∈[ ] ≥ ∈ ] ≥ be seen from the equation of X(t) that X(x,0)(t) β1, H if t tx, x (0, β1 . For t T3, we 1 t can use (4.8) and (4.9)to estimate t 0 ϒ(Z(x,0)(s))ds in the following three cases. Case 1. If t − T2 ≤ tx ≤ t then t tx t ϒ(Z(x,0)(s))ds = ϒ(Z(x,0)(s))ds + ϒ(Z(x,0)(s))ds 0 0 tx 7 3 MH ≥ γ0(t − T2) − T2MH ≥ γ0t, since t ≥ 4 + 7 T2 . 4 2 γ0 Case 2. If tx ≤ t − T2, then t tx t ϒ(Z(x,0)(s))ds = ϒ(Z(x,0)(s))ds + ϒ(Z(x,0)(s))ds 0 0 tx 7 7 3 ≥ γ (t − t ) + γ t ≥ γ t. 4 0 x 4 0 x 2 0 342 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 Case 3. If tx ≥ t, then t tx 7 3 ϒ(Z (s))ds = ϒ(Z (s))ds ≥ γ t ≥ γ t. (x,0) (x,0) 4 0 x 2 0 0 0 As a result, t 1 3 ϒ(Z (s))ds ≥ γ , if t ≥ T ,x∈ (0,H]. (4.10) t (x,0) 2 0 3 0 Let T = T1 ∨ T3. By the continuous dependence of solutions on initial values, there is β>0 such that T 1 1 X (T ) ∨ Y (T ) ≤ H and ϒ(Z (s)) − ϒ(Z (s)) ds ≤ γ (4.11) z z T z1 z2 2 0 0 2 given that |z1 − z2| ≤ β, z1, z2 ∈[0, H] . Combining (4.7), (4.10) and (4.11)we obtain the de- sired result. Generalizing the techniques in [33], we divide the proof of the eventual tightness into two lemmas. 2 Lemma 4.4. For any > 0, there exist ε0, δ0, T>0 and a compact set K ⊂ Int R+ such that k−1 1 ε,δ 2 lim inf P Z (nT ) ∈ K ≥ 1 − for any ε<ε0,δ<δ0,z∈ Int R+. k→∞ k z0,i0 3 n=0 Proof. For any > 0, let H = H( ) > a + 1be chosen later and define D ={(x, y) : 0 < b x, y ≤ H }. Let T>0 and β>0 such that (4.5)is satisfied and D1 ={(x, y) : 0 t % & δ 2c V(Zε,δ(t)) − V(z)= −ϒ Zε,δ(s), αε(s) + λ2(αε(s)) + ρ2(αε(s)) ds z,i z,i 2 a 0 t t 2c √ √ − δλ(αε(s))dW (s) − δρ(αε(s))dW (s). a 1 2 0 0 For A ∈ F, using Holder’s inequality and Itô’s isometry, we have 343 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 ε,δ − E 1A V(Zz,i (T )) V(z) T T ε,δ ε δ 2c 2 ε 2 ε ≤ E1A ϒ Z (t), α (t) dt + E1A λ (α (t)) + ρ (α (t)) dt z,i 2 a 0 0 T T √ √ 2c ε ε + E1A δλ(α (t))dW1(t) + E1A δρ(α (t))dW2(t) a 0 0 ⎛ ⎞ 1 (4.12) T % & 2 1 ⎝ ε,δ ε δ 2c 2 ε 2 ε ⎠ ≤T(E1 ) 2 E ϒ Z (t), α (t) + λ (α (t)) + ρ (α (t)) dt A z,i 2 a 0 ⎛ ⎞ 1 T 2 3 2c + δ P(A) ⎝E λ2(αε(t)) + ρ2(αε(t)) dt⎠ a 0 3 ≤K4T(1 +|z|) P(A), where the last inequality follows from (4.2) and the boundedness of ρ(i) and λ(i). If A = , we have 1 E V(Zε,δ(T )) − V(z) ≤ K (1 +|z|). (4.13) T z,i 4 ˆ ˆ 2 Let HT >H such that Xz(t) ∨ Y z(t) ≤ HT for all z ∈[0, H ] , 0 ≤ t ≤ T and $ ∂ϒ ∂ϒ 2 ˆ d = sup (x, y) , (x, y) : (x, y) ∈ R+,x∨ y ≤ H . H ∂x ∂y T Let ς>0. Lemma 2.2 implies that there are δ0, ε0 such that if ε<ε0, δ<δ0, | − ε,δ |+| − ε,δ | ∧ γ0 ∈[ ] − ς ∈ P Xz(t) Xz,i (t) Y z(t) Yz,i (t) < 1 , for all t 0,T > 1 ,z D. 2dH 6 (4.14) | − ε,δ | +| − ε,δ | ∧ γ0 On the other hand, if Xz(t) Xz,i (t) Y z(t) Yz,i (t) < 1 , we have 2dH T T 1 ε,δ ε 1 ϒ(Z (t), α (t))dt − ϒ(Zz,i(t))dt T z,i T 0 0 T 1 ε,δ ≤ ϒ(Z (t)) − ϒ(Zz,i(t)) dt T z,i 0 344 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 T 1 ε,δ ε ε,δ + ϒ(Z (t), α (t)) − ϒ(Z (t)) dt T z,i z,i 0 T γ0 FH ≤ + 1{αε(t)=j} − vj dt, (4.15) 2 T ∈M 0 j 2 where FH := sup{|ϒ(z, i)|i ∈ M, z ∈[0, KT + 1] }. In view of [24, Lemma 2.1], T T/ε 2 2 1 ε κ E 1{αε(t)=j} − vj dt = E 1{α(t)=j} − vj dt ≤ ε (4.16) T ∈M T ∈M T 0 j 0 j for some constant κ>0. On the one hand, T 2 2 1 2c ε ε 4c 2 2 E − λ(α (t))dW1(t) − ρ(α (t))dW2(t) ≤ λ + ρ . (4.17) T a a2 M M 0 Combining (4.5), (4.14), (4.15), (4.16), and (4.17), we can reselect ε0 and δ0 such that for ε<ε0, δ<δ0 we have ⎧ ⎫ ⎨ T ⎬ −1 ς P ϒ(αε(t), Zε,δ(t))dt ≤−0.5γ ≥ 1 − ,z∈ D ,i∈ M, (4.18) ⎩ T z,i 0⎭ 3 1 0 ς P Xε,δ(T ) ∨ Y ε,δ(T ) ≤ H (or equivalently Zε,δ(T ) ∈ D) ≥ 1 − ,z∈ D , (4.19) z,i z,i z,i 3 1 and ⎧ ⎫ √ T ⎨ ⎬ δ 2c ε ε ς P δϑ + λ(α (t))dW1(t) + ρ(α (t))dW2(t) dt < 0.25γ0 > 1 − (4.20) ⎩ T a ⎭ 3 0 = 1 2c 2 + 2 ∈ × M ε,δ ⊂ where ϑ 2 a λM ρM . Consequently, for any (z, i) D1 , there is a subset z,i ε,δ ≥ − ε,δ ∈ with P( z,i ) 1 ς in which we have Zz,i (T ) D and T 1 −1 V(Zε,δ(T )) − V(z) ≤ ϒ(αε(t), Zε,δ(t))dt + δϑ T z,i T z,i 0 T (4.21) 1 √ 2c + δ λ(αε(t))dW (t) + ρ(αε(t))dW (t) T a 1 2 0 ≤−0.25γ0 345 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 On the other hand, we deduce from (4.13) that for z ∈ D, 1 P V(Zε,δ(T )) − V(z) ≤ ! ≥ 1 − ς, (4.22) T z,i + := K4(1 2H) ∈ \ where ! ς . Moreover, it also follows from (4.13) that for z D D1 ε,δ ≤ + | | EV(Zz,i (T ) sup V(z) K4T z . z∈D\D1 Define L1 := sup V(z)+ !T , L2 := L1 + 0.25γ0, (4.23) z∈D\D1 2 as well as D2 := {(x, y) ∈ Int R+ : (x, y) ∈ D, V(x, y) >L2} and U(z) = V(z) ∨ L1. It is clear that 2◦ U(z2) − U(z1) ≤|V(z2) − V(z1)| for any z1,z2 ∈ R+ . (4.24) It follows from (4.12) that for any δ, ε<1, A ∈ F, and z ∈ D, we have 1 3 E1 V(Zε,δ(T )) − V(z) ≤ K (2H + 1) P(A). (4.25) T A z,i 4 = \ ε,δ Applying (4.25) and (4.24) with A z,i , we get " # √ 1 ε,δ − ≤ + ∈ E1 \ ε,δ U(Zz,i (T )) U(z) K4(2H 1) ς, if z D1. (4.26) T z,i ∈ ε,δ − In view of (4.21), for z D2 we have V Zz,i (T ) 346 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 ε,δ − ≤ U(Zz,i (T )) U(z) 0. This and (4.26)imply " # 1 √ EU(Zε,δ(T )) − U(z) ≤ K (2H + 1) ς, ∀z ∈ D \ D . (4.29) T z,i 4 1 2 If z ∈ D \ D1, U(z) = L1 and we have from (4.22) and (4.23) that ) * ) * ε,δ = = ε,δ ≤ ≥ − P U(Zz,i (T )) L1 P V(Zz,i (T )) L1 1 ς. Thus { ε,δ = }≥ − P U(Zz,i (T )) U(z) 1 ς. Use (4.25) and (4.24)again to arrive at " # 1 √ EU(Zε,δ(T )) − U(z) ≤ K (2H + 1) ς, ∀ z ∈ D \ D . (4.30) T z,i 4 1 On the other hand, equations (4.13) and (4.24)imply " # 1 ε,δ 2 EU(Z (T )) − U(z) ≤ K (1 +|z|), z ∈ Int R+. (4.31) T z,i 4 2 Pick an arbitrary (z0, i0) ∈ Int R+ × M. An application of the Markov property yields " # 1 EU(Zε,δ ((n + 1)T )) − EU(Zε,δ (nT )) T z0,i0 z0,i0 " # 1 = EU(Zε,δ(T )) − U(z) P Zε,δ (nT ) ∈ dz,αε(t) = i . T z,i z0,i0 i∈M 2 Int R+ Combining (4.28), (4.29), (4.30), and (4.31), we get " # 1 EU(Zε,δ ((n + 1)T )) − EU(Zε,δ (nT )) T z0,i0 z0,i0 √ ) * ≤− 0.25γ (1 − ς)− K (2H + 1) ς P Zε,δ (nT ) ∈ D 0 4 z0,i0 2 √ ) * + K (2H + 1) ςP Zε,δ (nT ) ∈ D \ D 4 z0,i0 2 ε,δ + K E1 ε,δ 1 +|Z (nT )| 4 {Z (nT )/∈D} z0,i0 z0,i0 ) * √ ≤−0.25γ (1 − ς)P Zε,δ (nT ) ∈ D + K (2H + 1) ς 0 z0,i0 2 4 ) * + K P Zε,δ (nT )∈ / D E 1 +|Zε,δ (nT )| . 4 z0,i0 z0,i0 347 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 Note that − " # 1 k 1 1 lim inf EU(Zε,δ ((n + 1)T )) − EU(Zε,δ (nT )) k→∞ k T z0,i0 z0,i0 n=0 1 = lim inf EU(Zε,δ (kT )) ≥ 0. k→∞ kT z0,i0 This forces k−1 ) * − 1 ε,δ ∈ 0.25γ0(1 ς)lim sup P Zz ,i (nT ) D2 →∞ k 0 0 k n=0 √ k ) * ≤ + + 1 ε,δ ∈ +| ε,δ | K4(2H 1) ς K3 lim sup P Zz ,i (nT ) / D E 1 Zz ,i (nT ) . →∞ k 0 0 0 0 k n=1 (4.32) In view of Lemma 4.1, we can choose H = H( )independent of (z0, i0) such that ε,δ ) * E(|Z (t)|) ε,δ ∈ ≤ z0,i0 ≤ lim sup P Zz ,i (t) / D lim sup , (4.33) t→∞ 0 0 t→∞ H 6 and ) * ε,δ ∈ +| ε,δ | K4 lim sup P Zz ,i (t) / D E 1 Zz ,i (t) t→∞ "0 0 # 0 0 2 E 1 +|Zε,δ (t)| z0,i0 0.1γ0 ≤ K4 lim sup ≤ . t→∞ H 6 Hence, we have k ) * 1 ε,δ ∈ +| ε,δ | ≤ 0.1γ0 K4 lim sup P Zz ,i (nT ) / D E 1 Zz ,i (nT ) . (4.34) →∞ k 0 0 0 0 6 k n=1 √ 0.1γ Choose ς = ς(H ) > 0 such that 0.25γ (1 − ς) ≥ 0.2γ and K (2H + 1) ς ≤ 0 and let 0 0 4 6 ε0 = ε0(ς, H), δ0(ς, H)such that (4.18), (4.19), and (4.20) hold. As a result, we get from (4.32) and (4.34) that k ) * 1 ε,δ ∈ ≤ lim sup P Zz ,i (nT ) D2 . (4.35) →∞ k 0 0 6 k n=1 This together with (4.33) and (4.35)shows that for any ε<ε0, δ<δ0, we have k ) * 1 ε,δ lim inf P Z (nT ) ∈ D \ D2 ≥ 1 − . (4.36) k→∞ k z0,i0 3 n=1 348 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 2 One can conclude the proof by noting that the set D \ D2 is a compact subset of Int R+. Lemma 4.5. There are L > 1, ε1 = ε( ) > 0, and δ1 = δ1( ) > 0 such that as long as 0 <ε< ε1, 0 <δ<δ1, we have T 1 ε,δ −1 2 2 lim inf P{Z (t) ∈[L ,L] }≥1 − , (z0,i0) ∈ Int R+ × M. T →∞ T z0,i0 0 2 Proof. Let D and T be as in Lemma 4.4. Since D \ D2 is a compact set in Int R+, by a mod- ification of the proof of [25, Theorem 2.1], we can show that there is a positive constant L > 1 such that P{Z (t) ∈[L−1, L]2} > 1 − , z ∈ D \ D , i ∈ M, 0 ≤ t ≤ T . Hence, it follows from z,i 3 2 the Markov property of the solution that − P{Zε,δ (t) ∈[L 1,L]2} z0,i0 ) * ε,δ ≥ 1 − P Z (jT ) ∈ D \ D2 ,t ∈[jT,jT + T ]. 3 z0,i0 Consequently, kT ) * 1 − lim inf P Zε,δ (t) ∈[L 1,L]2 dt k→∞ kT z0,i0 0 k−1 ) * 1 ε,δ ≥ 1 − lim inf P Z (jT ) ∈ D \ D2) ≥ 1 − . 3 k→∞ k z0,i0 j=0 It is readily seen from this estimate that T ) * 1 − lim inf P Zε,δ (t) ∈[L 1,L]2 dt ≥ 1 − . T →∞ T z0,i0 0 Theorem 1.2. Suppose Assumption 1.3 holds. For sufficiently small δ and ε, the process given ε,δ 2 2 by (1.9) has a unique invariant probability measure μ with support in Int R+ (where Int R+ 2 denotes the interior of R+). In addition: δ ε,δ a) If lim = l ∈ (0, ∞], the family of invariant probability measures (μ )ε>0 converges ε→0 ε weakly to μ0, the occupation measure of the limit cycle of (1.10), as ε → 0 (in the sense of Theorem 1.1). δ b) If lim = 0 and at each equilibrium (x∗, y∗) of (ϕ(x, y), ψ(x, y)), there is i∗ ∈ M such ε→0 ε that either ϕ(x∗, y∗, i∗) = 0 or ψ(x∗, y∗, i∗) = 0, then the family of invariant probability ε,δ 0 measures (μ )ε>0 converges weakly to μ , the occupation measure of the limit cycle of (1.10) as ε → 0. 349 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 Proof. The conclusion of Lemma 4.5 is sufficient for the existence of a unique invariant proba- ε,δ 2 ε,δ ε bility measure μ in Int R+ × M of (Z (t), α (t)) (see [4]or [31]). Moreover, the empirical measures t 1 ) * P Zε,δ (s) ∈· ds,t > 0 t z0,i0 0 converge weakly to the invariant probability measure με,δ as t →∞. Applying Fatou’s lemma to the above estimate yields ε,δ −1 2 μ ([L ,L] ) ≥ , ∀ ε<ε0,δ<δ0. This tightness implies Theorem 1.2. 4.1. An example In this section we provide a specific example under the setting of Section 4. We consider the following stochastic predator-prey model with Holling functional response in a switching regime ⎧ % & ε,δ ε ε,δ ε,δ ⎪ ε,δ ε ε,δ x (t) m(α (t))x (t)y (t) ⎪ dx (t) = r(α (t))x (t) 1 − − dt ⎪ K(αε(t) a(αε(t)) + b(αε(t))xε,δ(t) ⎪ √ ⎨⎪ ε ε,δ + δλ(α (t))x (t)dW1(t) % & (4.37) ⎪ ε ε ε,δ ⎪ ε,δ = ε,δ − ε + e(α (t))m(α (t))x (t) − ε ε,δ ⎪ dy (t) y (t) d(α (t)) f(α (t))y (t) dt ⎪ a(αε(t)) + b(αε(t))xε,δ(t) ⎩⎪ √ ε ε,δ + δρ(α (t))x (t)dW2(t), ε where W1 and W2 are two independent Brownian motions, α (t) is a Markov chain, that is independent of the Brownian motions, with state space M ={1, 2} and generator Q/ε where −11 Q = , (4.38) 1 −1 and r(1) = 0.9, r(2) = 1.1, K(1) = 4.737, K(2) = 5.238, m(1) = 1.2, m(2) = 0.8, a(1) = a(2) = 1, b(1) = b(2) = 1, d(1) = 0.85, d(2) = 1.15, e(1) = 1, e(2) = 2.5, f(1) = 0.03, f(2) = 0.01, λ(1) = 1, λ(2) = 2, ρ(1) = 3, ρ(2) = 1. As ε and δ tend to 0, solutions of equation (4.37) converge to the corresponding solutions of ⎧ ⎪ d x(t) x(t)y(t) ⎨⎪ x(t) = x(t) 1 − − , dt 5 1 + x(t) (4.39) ⎪ d 1.6x(t) ⎩⎪ y(t) = y(t) −1 + − 0.02y(t) dt 1 + x(t) 350 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 Fig. 1. From left to right: Graphs of the xε,δ(t) component of (4.37) with (ε, δ) = (0.001, 0.001), (ε, δ) = (0.00005, 0.00005) and x(t) of the averaged system (4.39) respectively. Fig. 2. From left to right: Graphs of the yε,δ(t) component of (4.37) with (ε, δ) = (0.001, 0.001), (ε, δ) = (0.00005, 0.00005) and y(t) of the averaged system (4.39), respectively. on any finite time interval [0, T ]. The system (4.39) has the unique equilibrium (x∗, y∗) = (1.836, 1.795). Modifying [35, Theorem 2.6] it can be seen that the solution of equation (4.39) has a unique limit cycle that attracts all positive solutions except for (x∗, y∗). Moreover, it is easy to check that the drift ⎛ ⎞ x(t) m(i)x(t)y(t) ⎜ r(i)x(t) 1 − − ⎟ ⎜ K(i) a(i) + b(i)x(t) ⎟ ⎜ ⎟ (4.40) ⎝ e(i)m(i)x(t) ⎠ y(t) −d(i)− − f(i)y(t) a(i) + b(i)x(t) does not vanish at (1.836, 1.795). The assumptions of Theorem 1.2 hold in this example. As a ε,δ result, the family (μ )ε>0 converges weakly as ε → 0to the stationary distribution of (4.39) that is concentrated on the limit cycle . We illustrate this convergence in Figs. 1, 2 and 3 by graphing sample paths of (4.37)for different values of (ε, δ). Fig. 4 provides the corresponding occupation measure. 351 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 Fig. 3. Phase portraits of (4.37)for different values of ε and δ. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.) Fig. 4. The two figures illustrate an approximate density of the occupation measure on the time interval [0, 1000] with ε = δ = 0.001 on the right and ε = δ = 0.0001 on the left. The step size is h = 0.0001. We divide the domain in 50 × 50 cells, approximate the density by the frequency of the process staying in each cell, and then interpolate. The simulations support the theoretical results that the occupation measures converge to the invariant probability measure. 4.2. Another example Consider the following system ⎧ ⎪dxε,δ(t) = −yε,δ(t) + xε,δ 1 − a(αε(t)) (xε,δ(t))2 + (yε,δ(t))2 (1 + (zε,δ(t))2) dt ⎪ ⎪ √ ⎪ + δdW1(t) ⎨ dyε,δ(t) = −xε,δ(t) + yε,δ 1 − a(αε(t)) ∗ (xε,δ(t))2 + (yε,δ(t))2 (1 + (zε,δ(t))2) ⎪ √ ⎪ ⎪ + δdW2(t) ⎪ √ ⎩ ε,δ ε,δ ε,δ 2 ε ε,δ 2 ε,δ 2 dz (t) = z (t) 1 − (z (t)) − b(α (t) (x (t)) + (y (t)) dt + δdW3(t) (4.41) 352 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 Fig. 5. The left figure is a sample path starting at the equilibrium (0, 0, 0) of the limit system. We see that this path has a tendency of approaching (0, 0, 1) before it is finally attracted to the proximity of the limit cycle. The right figure is an approximated density of the occupation measure of two components x, y on the time interval [0, 1000]. with a(1) = 0.8, a(2) = 1.2, b(1) = 3.5 and b(2) = 4.5 and Q defined as in (4.38). The limit system is ⎧ ⎪ dx =−+ − 2 − 2 + 2 ⎨⎪ dt y x(1 x y )(1 z ) dy = + − 2 − 2 + 2 (4.42) ⎪ dt x y(1 x y )(1 z ) ⎩⎪ dz = − 2 − 2 − 2 dt z(1 z 4x 4y ), which has a unique limit cycle {z = 0, x2 + y2 = 1} and three hyperbolic equilibria: (0, 0, 0) whose eigenvalues have both negative and positive parts and (0, 0, ±1) which are sources (Fig. 5). Appendix A. Proofs of lemmas in Section 2 Lemma A.1. For any R, T, γ>0, there exists a number k1 = k1(R, T, γ) > 0 such that for all sufficiently small δ, k P{|Xε,δ(t) − ξ ε (t)|≥γ, for some t ∈[0,T]} < exp − 1 ,x ∈ B , x,i x,i δ R ε,δ ε where Xx,i(t) and ξx,i(t) are the solutions to the systems (1.3) and (1.7) that have initial value (x, i). Proof. By (i) and (ii) of Assumption 1.1, we can deduce the existence and boundedness of a unique solution to equation (1.7)using the Lyapunov function method. Moreover, we can find RT >R>0 such that almost surely 353 N.H. Du, A. Hening, D.H. Nguyen et al. Journal of Differential Equations 293 (2021) 313–358 | ε | − ∈[ ] ∈ ξx,i(t)